I agree with some aspects of Skiddles' answer, but not all.
Assume your data set contains n observations. Based on your question, I see three possibilities:
If you're interested in the number of successes (1s) in n trials, then your standard deviation should be sqrt(np(1-p)), which is the standard deviation for a binomial distribution.
If you're interested in a specific ORDER of successes and failures, then you should be using sqrt(p(1-p)), which is the standard deviation for a Bernoulli distribution.
If you're interested in estimating how the sample proportion of successes will change if you get additional samples, then you should sqrt(p(1-p))/n), which is the standard deviation of the sampling distribution of the sample proportion.
As to what value of p to use, if you don't know, then it's typical to use the most conservative value, 0.5, and conduct a hypothesis test to determine if the actual proportion is different from the hypothesized proportion of 0.5. That would be a Z test.
EDIT based on comment:
The question was "is the historical value of p the right value to use for calculating the standard deviation"?
Answer: No, I think you're confusing two separate but related concepts. You'll often see binomial distribution-type questions saying things like "based on past studies, it is assumed that the probability of success is 0.4" or something like that. However, this is not a case of "past studies", as far as I can see. It looks like you have only one study, in which case you don't really know the population proportion (p), you only have the sample proportion (p hat). You should use 0.5 for p, to calculate the standard deviation.